Rank and order of a finite group admitting a Frobenius-like group of automorphisms

A finite group $FH$ is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup $F$ with a nontrivial complement $H$ such that $FH/[F,F]$ is a Frobenius group with Frobenius kernel $F/[F,F]$. Suppose that a finite group $G$ admits a Frobenius-like group of automorphisms $FH$ of coprime order with certain additional restrictions (which are satisfied, in particular, if either $|FH|$ is odd or $|H|=2$). In the case where $G$ is a finite $p$-group such that $G=[G,F]$ it is proved that the rank of $G$ is bounded above in terms of $|H|$ and the rank of the fixed-point subgroup $C_G(H)$, and that $|G|$ is bounded above in terms of $|H|$ and $|C_G(H)|$. As a corollary, in the case where $G$ is an arbitrary finite group estimates are obtained of the form $|G|\le|C_G(F)|\cdot f(|H|,|C_G(H)|)$ for the order, and $\mathbf r(G)\le\mathbf r(C_G(F))+g(|H|,\mathbf r(C_G(H)))$ for the rank, where f and g are some functions of two variables.